Atlas › Modular forms › Weight 4 › Level 165 › 165.4.a › 165.4.a.b

Newform orbit 165.4.a.b

$q$ -expansion

$q + q^{2} + 3q^{3} - 7q^{4} - 5q^{5} + 3q^{6} + 36q^{7} - 15q^{8} + 9q^{9} - 5q^{10} + 11q^{11} - 21q^{12} + 2q^{13} + 36q^{14} - 15q^{15} + 41q^{16} + 66q^{17} + 9q^{18} + 140q^{19} + 35q^{20} + 108q^{21} + 11q^{22} - 68q^{23} - 45q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{7}$ is a root of

$x - 109$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$+1$
11	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	1	3	−5	36	11	2	66	140	−68	150	−128	−314	−118	172	−324	82	−740	122	−124	−988	2	1100	−868	−470	1186

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 165, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.15 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 3, 5, 11, 15, 33, 55, 165]: [1, 2, 3, 4, 8, 14, 20, 76] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(165, 4)[1]   # 165.4.a.b

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 165:4 1000